No abstract

We introduce the notion of a Seshadri stratification on an embedded projective variety. Such a structure enables us to construct a Newton-Okounkov simplicial complex and a flat degeneration of the projective variety into a union of toric varieties. We show that the Seshadri stratification provides a geometric setup for a standard monomial theory. In this framework, Lakshmibai-Seshadri paths for Schubert varieties get a geometric interpretation as successive vanishing orders of regular functions.

Abstract. Let σ be a simple involution of an algebraic semisimple group G and let H be the subgroup of G of points fixed by σ. If the restricted root system is of type A, C or BC and G is simply connected or if the restricted root system is of type B and G is adjoint, then we describe a standard monomial theory and the equations for the coordinate ring k[G/H] using the standard monomial theory and the Plücker relations of an appropriate (maybe infinite dimensional) Grassmann variety.The aim of this paper is the description of the coordinate ring of the symmetric varieties and of certain rings related to their wonderful compactification. The main tool to achieve this goal is a (possibly infinite dimensional) Grassmann variety associated to a pair consisting of a symmetric space and a spherical representation.More precisely, let G be a semisimple algebraic group over an algebraically closed field k of characteristic 0 and let σ be a simple involution of G (i.e. G ⋊{id, σ} acts irreducibly on the Lie algebra of G). Let H = G σ be the fixed point subgroup. Fix a spherical dominant weight ε in Ω + . We add a node n 0 to the Dynkin diagram of G and, for all simple roots α, we join n 0 with the node n α of the simple root α by ε(α ∨ ) lines, and we put an arrow in direction of n α if ε(α ∨ ) ≥ 2. In the cases relevant for us, the Kac-Moody group e G associated to the extended diagram will be of finite or affine type. Let L be the ample generator of Pic(Gr) for the generalized Grassmann varietyP . The homogeneous coordinate ring Γ Gr = j≥0 Γ(Gr, L j ) is the quotient of the symmetric algebra S(Γ(Gr, L)) by an ideal generated by quadratic relations, the generalized Plücker relations.Since our aim is to relate these Plücker relations to k[G/H], we say that the monoid Ω + is quadratic if (it is free and) its basis has the following property with respect to the dominant order of the restricted root system: any element of Ω + that is less than the

We study the ring of sections A (X) of a complete symmetric variety X, that is of the wonderful completion of G/H where G is an adjoint semisimple group and H is the fixed subgroup for an involutorial. automorphism of G. We find generators for Pic(X), we generalize the PRV conjecture to complete symmetric varieties and construct a standard monomial theory for A(X) that is compatible with G orbit closures in X. This gives a degeneration result and the rational singularityness for A (X)

Abstract.A general theory of LS algebras over a multiposet is developed. As a main result, the existence of a flat deformation to discrete algebras is obtained. This is applied to the multicone over partial flag varieties for Kac-Moody groups proving a deformation theorem to a union of toric varieties. In order to achieve the Cohen-Macaulayness of the multicone we show that Bruhat posets (defined as glueing of minimal representatives modulo parabolic subgroups of a Weyl group) are lexicographically shellable. Mathematics Subject Classification (2000). 14M15, 06A07.